Iron, cobalt, nickel and some other materials and alloys constitute a special group of substances by the nature of the reaction to the action of a magnetic field. In an external relatively weak magnetic field, they are able to acquire a large magnetization. These metals are united by the name of ferromagnets, and the phenomena associated with their magnetization are called ferromagnetism. Ferromagnets do not have a linear relationship between magnetic induction and field strength. Magnetic permeability and magnetic susceptibility. Magnetization and magnetization reversal of ferromagnets

do not have constant value. Wherein

Ferromagnets have a crystalline structure. From experience it is known that their magnetization is associated only with the orientation of the spins of electrons. Moreover, in the structure of matter there are separate regions with a predominant direction of spins, called regions of spontaneous magnetization or magnetic domains. In the absence of an external magnetic field, the magnetic moments of the domains are mutually balanced so that the total magnetization of the substance is zero.

However, sometimes even a very weak external field is enough for this equilibrium to be violated and the magnetic moments of some domains to be oriented along the external magnetic field, and the ferromagnet to be highly magnetized. As the intensity increases, the magnetization increases until a state is reached in which the magnetic moments of all domains are oriented in the direction of the external field (point m in Fig. 2.1, a). This condition is called magnetic saturation. A further increase in the field strength will no longer affect the magnetization of the substance.

Fig. 2.1

The shape of the curve () for a single crystal of a ferromagnet depends on the direction of the external field with respect to the axes of the crystal. The crystal is easiest to magnetize along the edges of the crystal lattice (curve a in Fig. 2.1, a). It is most difficult along the large diagonal of the crystal (curve c in Fig. 2.1, a), and some average magnetization is achieved in the direction of the diagonals of the crystal faces (curve b in Fig. 2.1, a). JH Fig. 2.1 Using expression (1.16) and curve (), we can construct the dependence of the magnetic field induction in a ferromagnet on the intensity – JH () B H.

It is the sum of the ordinates of the curve () and the straight line of induction in vacuum JH 0B H = μ. Since const J = in the saturation region, the characteristic () B H in this region becomes linear with a slope corresponding to the magnetic constant (Fig. 2.1, b). 0μ Each point of the curve () B H in Fig. 2.1, b corresponds to the value of magnetic permeability. In the initial section, with a weak field, it is relatively small, and then increases sharply, reaching a maximum. After that, permeability decreases as it saturates.

In addition to absolute magnetic permeability, the state of a ferromagnet with small changes in the field strength is characterized by differential magnetic permeability / tga BH μ = α αμ tgdB dH = β, which is the tangent of the angle of inclination of the tangent to the magnetization curve. As can be seen from fig. 2.1, b, it may differ significantly from. aμ If, upon reaching saturation, the field strength H is gradually reduced, the magnetic moments of individual domains will return to their original state.

However, when the external field 0 H = disappears, the ferromagnet retains some residual magnetization r JB r = (Fig. 2.2). To reduce the induction to zero, it is necessary to create an external magnetic field with the opposite direction and intensity c H−, called the coercive or restraining force. Further increase tension reverse field will lead to a saturation state of the ferromagnet with the opposite direction of induction.

With a cyclic change in the external field from s H + to s H−, the magnetization and induction of the ferromagnet will change along a curve that has a closed shape and is called a hysteresis loop. The term comes from the buckwheat υστερησιζ – lag, and is associated with the fact that the change in the magnetic field induction occurs with delay relative to the change in intensity. This leads to the fact that the magnitude of the induction is determined not only by the value of the external field strength at a given moment in time, but also by the previous value induction.

Fig. 2.2

The hysteresis loop is characterized by the values of maximum and residual induction s B and c B, as well as the coercive force c H. If the magnetization reversal is performed to the saturation state, then the corresponding curve is called the limit hysteresis loop (Fig. 2.2).

In the case of symmetric magnetization reversal, i.e. when the maximum values of the magnetic field in both directions are the same, the curves () B H form loops inside the limit curve (Fig. 2.3, a). In this case, the vertices of the symmetric cycles are located on a curve called the main magnetization curve.

Fig. 2.3

If, after bringing the ferromagnet to saturation, the tension is reduced, and then, starting from some point A, it starts to increase, then magnetic induction will increase according to the so-called return curve, approaching the main magnetization curve but without crossing it (Fig. 2.3, b).

1.2.2. Magnetic field energy

A magnetic field contains energy distributed over the space occupied by the field. It corresponds to the work expended by an electric current when creating a field.

Fig. 2.4

Let the magnetic field be created by the current flowing through the winding of the toroidal coil with the number of turns w and resistance R (Fig. 2.4). When the winding is connected to a power source, the current in it will not be established immediately, but gradually, because magnetic flux covering windings induces self-induction EMF

which prevents the rise of current. The Kirchhoff equation for the electrical circuit of the winding has the form

The energy coming from the power source dWq in time dt is

Part of this energy is dissipated as heat in the resistance of the winding, and another part

if no work is done, it increases the energy of the magnetic field. The magnetic field strength on the axis of the coil in accordance with (1.18) is equal to / H iw l =.

Therefore, the current strength can be represented as. At the same time, the flux linkage of the toroidal coil is S i = H l /w ,Ψ = wΦ = wBs. Substituting expressions for flux linkage and current in (2.1), we find the magnetic field energy in the form

If both parts are divided by V, then we obtain an expression for the energy density, which is also true for an inhomogeneous magnetic field, because any field within infinitesimal limits can be considered as homogeneous, i.e. B

In the event that the magnetic permeability of the medium at all points of the magnetic field is the same (const aμ =), then the energy density is

When the magnetic field energy can also be expressed in terms of inductance

where is the final value of the current. This allows us to determine the inductance as a specific measure of the energy of the magnetic field, because it is equal to its double value at a unit current.